Expression of type ExprTuple

from the theory of proveit.physics.quantum.circuits

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import A, Conditional, ExprRange, ExprTuple, IndexedVar, Lambda, N, Variable, VertExprArray, m, n
from proveit.core_expr_types import A_1_to_m
from proveit.linear_algebra import TensorProd
from proveit.logic import And, InSet
from proveit.numbers import Add, Interval, Natural, one, subtract, zero
from proveit.physics.quantum.circuits import Input, MultiQubitElem, N_0_to_m, N_m, Qcircuit, QcircuitEquiv, each_Nk_is_partial_sum

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
sub_expr2 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr3 = Variable("_c", latex_format = r"{_{-}c}")
expr = ExprTuple(Lambda([N_0_to_m], Conditional(QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, ExprRange(sub_expr2, MultiQubitElem(element = Input(state = IndexedVar(A, sub_expr1), part = sub_expr2), targets = Interval(Add(IndexedVar(N, subtract(sub_expr1, one)), one), IndexedVar(N, sub_expr1))), one, IndexedVar(n, sub_expr1)).with_wrapping_at(2,6), one, m)])), Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr3, ExprRange(sub_expr1, MultiQubitElem(element = Input(state = TensorProd(A_1_to_m), part = sub_expr1), targets = Interval(one, N_m)), Add(IndexedVar(N, subtract(sub_expr3, one)), one), IndexedVar(N, sub_expr3)).with_wrapping_at(2,6), one, m)]))), And(ExprRange(sub_expr2, InSet(IndexedVar(N, sub_expr2), Natural), zero, m), each_Nk_is_partial_sum))))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left(\left(N_{0}, N_{1}, \ldots, N_{m}\right) \mapsto \left\{\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{A_{1}~\mbox{part}~1~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} & \qw \\
\qin{A_{1}~\mbox{part}~2~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{1}~\mbox{part}~n_{1}~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} & \qw \\
\qin{A_{2}~\mbox{part}~1~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} & \qw \\
\qin{A_{2}~\mbox{part}~2~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{2}~\mbox{part}~n_{2}~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} & \qw \\
\qin{\begin{array}{c}\vdots\\ \vdots\end{array}} & \qw \\
\qin{A_{m}~\mbox{part}~1~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{m}~\mbox{part}~2~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{m}~\mbox{part}~n_{m}~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}} & \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1 - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1 - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2 - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2 - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\begin{array}{c}\vdots\\ \vdots\end{array}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw
} \end{array}\right) \textrm{ if } \left(N_{0} \in \mathbb{N}\right) ,  \left(N_{1} \in \mathbb{N}\right) ,  \ldots ,  \left(N_{m} \in \mathbb{N}\right) ,  \left(N_{0} = 0\right)\land \left(N_{1} = \left(N_{1 - 1} + n_{1}\right)\right) \land  \left(N_{2} = \left(N_{2 - 1} + n_{2}\right)\right) \land  \ldots \land  \left(N_{m} = \left(N_{m - 1} + n_{m}\right)\right)\right..\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	ExprTuple	1
1	Lambda	parameters: 2 body: 3
2	ExprTuple	4
3	Conditional	value: 5 condition: 6
4	ExprRange	lambda_map: 7 start_index: 47 end_index: 94
5	Operation	operator: 8 operands: 9
6	Operation	operator: 19 operands: 10
7	Lambda	parameter: 103 body: 48
8	Literal
9	ExprTuple	11, 12
10	ExprTuple	13, 14
11	Operation	operator: 16 operand: 21
12	Operation	operator: 16 operand: 22
13	ExprRange	lambda_map: 18 start_index: 47 end_index: 94
14	Operation	operator: 19 operands: 20
15	ExprTuple	21
16	Literal
17	ExprTuple	22
18	Lambda	parameter: 103 body: 23
19	Literal
20	ExprTuple	24, 25
21	ExprTuple	26
22	ExprTuple	27
23	Operation	operator: 28 operands: 29
24	Operation	operator: 40 operands: 30
25	ExprRange	lambda_map: 31 start_index: 106 end_index: 94
26	ExprRange	lambda_map: 32 start_index: 106 end_index: 94
27	ExprRange	lambda_map: 33 start_index: 106 end_index: 94
28	Literal
29	ExprTuple	48, 34
30	ExprTuple	35, 47
31	Lambda	parameter: 103 body: 36
32	Lambda	parameter: 101 body: 37
33	Lambda	parameter: 88 body: 38
34	Literal
35	IndexedVar	variable: 91 index: 47
36	Operation	operator: 40 operands: 41
37	ExprRange	lambda_map: 42 start_index: 106 end_index: 43
38	ExprRange	lambda_map: 44 start_index: 45 end_index: 46
39	ExprTuple	47
40	Literal
41	ExprTuple	48, 49
42	Lambda	parameter: 103 body: 50
43	IndexedVar	variable: 67 index: 101
44	Lambda	parameter: 101 body: 51
45	Operation	operator: 97 operands: 52
46	IndexedVar	variable: 91 index: 88
47	Literal
48	IndexedVar	variable: 91 index: 103
49	Operation	operator: 97 operands: 54
50	Operation	operator: 56 operands: 55
51	Operation	operator: 56 operands: 57
52	ExprTuple	58, 106
53	ExprTuple	88
54	ExprTuple	59, 60
55	NamedExprs	element: 61 targets: 62
56	Literal
57	NamedExprs	element: 63 targets: 64
58	IndexedVar	variable: 91 index: 74
59	IndexedVar	variable: 91 index: 75
60	IndexedVar	variable: 67 index: 103
61	Operation	operator: 70 operands: 68
62	Operation	operator: 72 operands: 69
63	Operation	operator: 70 operands: 71
64	Operation	operator: 72 operands: 73
65	ExprTuple	74
66	ExprTuple	75
67	Variable
68	NamedExprs	state: 76 part: 103
69	ExprTuple	77, 78
70	Literal
71	NamedExprs	state: 79 part: 101
72	Literal
73	ExprTuple	106, 80
74	Operation	operator: 97 operands: 81
75	Operation	operator: 97 operands: 82
76	IndexedVar	variable: 99 index: 101
77	Operation	operator: 97 operands: 83
78	IndexedVar	variable: 91 index: 101
79	Operation	operator: 85 operands: 86
80	IndexedVar	variable: 91 index: 94
81	ExprTuple	88, 102
82	ExprTuple	103, 102
83	ExprTuple	89, 106
84	ExprTuple	101
85	Literal
86	ExprTuple	90
87	ExprTuple	94
88	Variable
89	IndexedVar	variable: 91 index: 95
90	ExprRange	lambda_map: 93 start_index: 106 end_index: 94
91	Variable
92	ExprTuple	95
93	Lambda	parameter: 103 body: 96
94	Variable
95	Operation	operator: 97 operands: 98
96	IndexedVar	variable: 99 index: 103
97	Literal
98	ExprTuple	101, 102
99	Variable
100	ExprTuple	103
101	Variable
102	Operation	operator: 104 operand: 106
103	Variable
104	Literal
105	ExprTuple	106
106	Literal